SC · Fourier

Introduction to Bode Plots

A Bode plot draws magnitude in decibels and frequency on a logarithmic axis. The once-curved frequency response then simplifies into straight-line asymptotes that bend at a corner, and see for yourself how those lines move along as a whole when the corner frequency moves.

Land the bend on the target corner

The upper figure is the Bode magnitude plot, with magnitude in decibels: the blue curve is the actual response and the gold straight lines are the asymptotes. The lower figure is the phase plot. Move the corner frequency ωc to bring the bend onto the dashed target ωc = 4, and thanks to the log axis a straight line of −20 dB/dec appears above the corner, falling 20 dB for every 10 times the frequency.

magnitude (dB)asymptotes

phase (deg)

corner frequency ωc1.00

Corner and asymptote

ωc = 1.00   −20 dB / dec

The bend is far from the target corner. Below the corner the line is a flat 0 dB, above it a straight line falling at −20 dB/dec, and the log axis is what straightens that attenuation into a line.

Corner off

Decibels and the log axis

A Bode plot draws magnitude in decibels, 20 log₁₀|H|, and frequency on a logarithmic axis. With logarithms, products become sums, so the decibel magnitudes of several stages connected in series simply add. A wide frequency range, from 0.1 to 1000 for example, also fits evenly into one picture, so a response that was multiplicative and sprawling moves into a world of addition and straight lines.

ObservedB = 20 log₁₀|H|

Decibels are 20 log₁₀|H|.

Choosefirst-order slope = ?

The first-order slope is −20 dB/dec.

Fill inat ωc: |H| = ?

At the corner the true magnitude is −3 dB.

On your ownat ωc: phase = ?

The phase at the corner is −45°.

The curve becomes straight lines

Drawn in dB on a log axis, the magnitude of a first-order lowpass straightens into two lines. Below the corner ωc it is a horizontal line at 0 dB, and above it a line of −20 dB/dec, falling 20 dB for every 10 times the frequency. The two lines meet at the corner and the actual curve is 3 dB below them there, so knowing just a few corner frequencies lets you sketch the whole response in straight pieces. For a second-order system the slope is −40 dB/dec, twice as steep.

Read the phase too

A Bode plot comes as a set of two, magnitude and phase. The phase of a first-order lowpass is 0° at frequencies well below the corner, −45° at the corner and −90° well above it, falling smoothly over about two decades centered on the corner. The corner in magnitude and the −45° in phase happen at the same ωc, so which frequencies a system cuts and delays, and by how much, can be read at a glance from the two pictures. The next unit reads bandwidth and filter type directly from this magnitude plot.

Back to the first screen

Each time the corner frequency moved, the bend slid along the log axis. Below the corner was always a flat 0 dB, above it always a straight line at −20 dB/dec, the two met at the corner, and the actual curve sat just 3 dB below there. The phase bent from 0° to −90° around the same corner. The power of the Bode plot lies in turning a once-curved frequency response into a few corners and straight pieces, so that even a wide frequency range can be drawn by hand.

A Bode plot draws magnitude in decibels, 20 log₁₀|H(ω)|, and phase in degrees (°), both on a logarithmic frequency axis, straightening the response into straight-line asymptotes. A first-order lowpass is 0 dB below the corner ωc and −20 dB/dec above it, its actual value at the corner is −3 dB, and its phase goes 0° → −45° at the corner → −90°. Because the log axis turns products into sums, the dB values of series stages add, so a few corner frequencies sketch the whole response, and an n-th-order system has a slope of −20n dB/dec.

On to the next unit

Now that you can read a Bode magnitude plot, it is time to pull practical numbers out of it. In the next unit, bandwidth and filtering, the width of the passband, the bandwidth up to −3 dB, is read straight from the corner, and four filters, lowpass, highpass, bandpass and bandstop, are told apart by the shape of their magnitude plots. Multiplying a signal's spectrum by a filter's response decides which frequencies survive, and here the property from C4 that convolution is multiplication becomes a hands-on tool for filter design.